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Santa's Dice Game

Q: Santa offers you to play a game of dice. You get to roll a dice six times. You can stop rolling whenever you wish and you get the dollar amount shown on that roll. What is an optimal strategy to maximize your payoff?

A: Let us take a moment and think through this. At each point in the sequence of rolls you make, you have a decision to make. Do you keep rolling or do you stop and walk away with what is being "offered" to you? You also need to bear in mind that if you keep pushing your luck you will reach a point (the 6th roll) where you would have to be content with whatever comes out for the last roll. So lets start with the simple case of what the expected payoff is for the last roll. Lets call this \(E_{6}\). To compute it, simply take the payoff multiplied by the respective probability.
$$
E_{6} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{7}{2} = 3.5
$$
The general strategy to be followed is to check what the expected pay off is for the next roll and accept nothing less than that. Working backwards, lets consider each of the rolls starting with the 5th roll.

Roll 5:
You know that on 6th roll you get an expected payoff of 3.5. You should accept anything greater than 3.5, which means you stop rolling if you get a 4,5 or 6, put in other words anything greater than 4. The expected payoff is
$$
\frac{4 + 5 + 6}{6} + \frac{7}{2}\times\frac{1}{2} = \frac{17}{4}
$$ Roll 4:
Looking ahead you find the expected payoff for the 5th roll is \(\frac{17}{4} = 4.25\). You stop if you get a 5 or greater for this roll. The expected payoff would be
$$
\frac{5+6}{6} + \frac{17}{4}\times\frac{4}{6} = \frac{14}{3}
$$

Roll 3:
The expected payoff from roll 4 above is \(\frac{14}{3} = 4.66\) which means you stop if you get a 5 or above. The expected payoff for this roll is
$$
\frac{5 + 6}{6} + \frac{14}{3}\times\frac{4}{6} = \frac{89}{18}
$$

Roll 1:
Being the first roll, your expected payoff from any subsequent rolls is 5.12. So you should accept nothing short of a 6 from the first roll.

The optimal strategy would then be to stop at each of the first five rolls (when you have the option to stop) if you get 6,5,5,5,4 respectively. An interesting takeaway here is the expected gains from following this strategy can be computed as
$$
\frac{1}{6}\times 6 + \frac{5}{6}\times 5.12 = 5.26
$$
which is quite high for a seemingly random game!

If you are looking to buy some books in probability here are some of the best books to own

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.